Classical phase space and statistical mechanics of identical particles (original) (raw)

Quantum particles from classical statistics

2009

Quantum particles and classical particles are described in a common setting of classical statistical physics. The property of a particle being "classical" or "quantum" ceases to be a basic conceptual difference. The dynamics differs, however, between quantum and classical particles. We describe position, motion and correlations of a quantum particle in terms of observables in a classical statistical ensemble. On the other side, we also construct explicitly the quantum formalism with wave function and Hamiltonian for classical particles. For a suitable time evolution of the classical probabilities and a suitable choice of observables all features of a quantum particle in a potential can be derived from classical statistics, including interference and tunneling. Besides conceptual advances, the treatment of classical and quantum particles in a common formalism could lead to interesting cross-fertilization between classical statistics and quantum physics.

Statistical mechanics of classical systems with distinguishable particles

2002

The properties of classical models of distinguishable particles are shown to be identical to those of a corresponding system of indistinguishable particles without the need for ad hoc corrections. An alternative to the usual definition of the entropy is proposed. The new definition in terms of the logarithm of the probability distribution of the thermodynamic variables is shown to be consistent with all desired properties of the entropy and the physical properties of thermodynamic systems. The factor of 1/N! in the entropy connected with Gibbs' Paradox is shown to arise naturally for both distinguishable and indistinguishable particles. These results have direct application to computer simulations of classical systems, which always use distinguishable particles. Such simulations should be compared directly to experiment (in the classical regime) without ''correcting'' them to account for indistinguishability.

Classical particles and order statistics

Physics Letters, 1989

It is shown that classical particles obey order statistics. This resolves the paradox of having to write in by hand the 1/Ni term in the N-particle partition function: This factor arises precisely because the particles are distinguishable rather than because there are Ni ways of permuting among themselves N particles which are physically indistinguishable and must be counted only once. Recently, the old and unsettled problem of whether dynamic properties, such as extensivity. The factor classical and quantum particles are distinguishable N! which must be introduced to achieve this, is noror not has been brought to the foreground. If quan-mally interpreted as indicating that classical partiturn statistics is a mere variation of classical statis-des are really indistinguishable.

Distinguishability or indistinguishability in classical and quantum statistics

Physics Letters, 1987

Contrary to a recent formulation ofclassical and quantum statistics in terms ofindistinguishable particles it is claimed that the notion of distinguishability can account for classical as well as for quantum statistics. The resulting non-local correlations are a manifestation ofa fluctuating stochastic metric which has been shown to produce the quantum potential.

General approach to quantum mechanics as a statistical theory

Physical review, 2019

Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 and Phys. Rev. A 96, 022117] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function-the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics-extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. Firstly we provide a brief modernized discussion of the general framework of phasespace quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space-putting it for the first time on a truly equal footing to Schrödinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of "parity" and "displacement" in a natural broadening of previously developed phase space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.

Quantum mechanics from classical statistics

2009

Quantum mechanics can emerge from classical statistics. A typical quantum system describes an isolated subsystem of a classical statistical ensemble with infinitely many classical states. The state of this subsystem can be characterized by only a few probabilistic observables. Their expectation values define a density matrix if they obey a "purity constraint". Then all the usual laws of quantum mechanics follow, including Heisenberg's uncertainty relation, entanglement and a violation of Bell's inequalities. No concepts beyond classical statistics are needed for quantum physics -the differences are only apparent and result from the particularities of those classical statistical systems which admit a quantum mechanical description. Born's rule for quantum mechanical probabilities follows from the probability concept for a classical statistical ensemble. In particular, we show how the noncommuting properties of quantum operators are associated to the use of conditional probabilities within the classical system, and how a unitary time evolution reflects the isolation of the subsystem. As an illustration, we discuss a classical statistical implementation of a quantum computer.

Statistics, Symmetry, and the Conventionality of Indistinguishability in Quantum Mechanics

Foundations of Phyiscs, 2000

The question to be addressed is, In what sense and to what extent do quantum statistics for, and the standard formal quantum-mechanical description of, systems of many identical particles entail that identical quantum particles are indistinguishable? This paper argues that whether or not we consider identical quantum particles as indistinguishable is a matter of theory choice underdetermined by logic and experiment.

Statistical approach to quantum mechanics I: General nonrelativistic theory

In this initial paper in a series, we first discuss why classical motions of small particles should be treated statistically. Then we show that any attempted statistical description of any nonrelativistic classical system inevitably yields the multi-coordinate Schr\"odinger equation, with its usual boundary conditions, as an essential statistical equation for the system. We derive the general "canonical quantization" rule, that the Hamiltonian operator must be the classical Hamiltonian in the NNN-dimensional metric configuration space defined by the classical kinetic energy of the system, with the classical conjugate momentum NNN-vector replaced by −ihbar-i\hbar−ihbar times the vector gradient operator in that space. We obtain these results by using conservation of probability, general tensor calculus, the Madelung transform, the Ehrenfest theorem and/or the Hamilton-Jacobi equation, and comparison with results for the charged harmonic oscillator in stochastic electrodynamics. ...

On the explanation for quantum statistics

Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 2006

The concept of classical indistinguishability is analyzed and defended against a number of well-known criticisms, with particular attention to the Gibbs'paradox. Granted that it is as much at home in classical as in quantum statistical mechanics, the question arises as to why indistinguishability, in quantum mechanics but not in classical mechanics, forces a change in statistics. The answer, illustrated with simple examples, is that the equilibrium measure on classical phase space is continuous, whilst on Hilbert space it is discrete. The relevance of names, or equivalently, properties stable in time that can be used as names, is also discussed.

ELECTROMAGNETIC EXCITATIONS OF A n QUANTUM HALL DROPLETS

International Journal of Modern Physics A, 2010

The classical description of A n internal degrees of freedom is given by making use of the Fock-Bargmann analytical realization. The symplectic deformation of phase space, including the internal degrees of freedom, is discussed. We show that the Moser's lemma provides a mapping to eliminate the fluctuations of the symplectic structure, which become encoded in the Hamiltonian of the system. We discuss the relation between Moser and Seiberg-Witten maps. One physics applications of this result is the electromagnetic excitation of a large collection of particles, obeying the generalized A n statistics, living in the complex projective space CP k with U (1) background magnetic field. We explicitly calculate the bulk and edge actions. Some particular symplectic deformations are also considered. Quantum Hall effect in higher dimensions has intensively been investigated in the last decade from different point of views . This yielded interesting results such as the bosonization [2, 11-13] that has been achieved by making use of the incompressible Hall droplet picture. In this framework, the edge excitations of a quantum Hall droplet are described by a generalized Wess-Zumino-Witten action. Recently, the electromagnetic excitations of a quantum Hall droplet was discussed by Karabali [11] and Nair for the Landau systems in the complex projective space CP k . In fact, the corresponding bulk and edge actions were derived . Interestingly, it was shown that the bulk contribution coincides with the (2k + 1)-dimensional Chern-Simons action .